# Answer to Question #22732 in Abstract Algebra for Irvin

Question #22732

If R is a commutative ring or a left noetherian ring, show that any finitely generated artinian left R-module M has finite length.

Expert's answer

If

*R*is left noetherian, then*M*is a noetherian (as well as artinian) module, so it has finite length.Now assume*R*is commutative. Since*M*is a finite sum of cyclicartinian submodules, we may assume*M*itself is cyclic. Represent*M*inthe form*R/I*, where*I**⊆**R*is a leftideal. Since*R*is commutative,*I*is an ideal, and the fact that*M*is artinian implies that*R/I*is an artinian ring. By theHopkins–Levitzki theorem,*R/I*is also a noetherian ring, so we are backto the case considered above.
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